The Stacks project

88.7 Formal objects and completion categories

In this section we discuss how to go between categories cofibred in groupoids over $\mathcal{C}_\Lambda $ to categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda $ and vice versa.

Definition 88.7.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. The category $\widehat{\mathcal{F}}$ of formal objects of $\mathcal{F}$ is the category with the following objects and morphisms.

  1. A formal object $\xi = (R, \xi _ n, f_ n)$ of $\mathcal{F}$ consists of an object $R$ of $\widehat{\mathcal{C}}_\Lambda $, and a collection indexed by $n \in \mathbf{N}$ of objects $\xi _ n$ of $\mathcal{F}(R/\mathfrak m_ R^ n)$ and morphisms $f_ n : \xi _{n + 1} \to \xi _ n$ lying over the projection $R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n$.

  2. Let $\xi = (R, \xi _ n, f_ n)$ and $\eta = (S, \eta _ n, g_ n)$ be formal objects of $\mathcal{F}$. A morphism $a : \xi \to \eta $ of formal objects consists of a map $a_0 : R \to S$ in $\widehat{\mathcal{C}}_\Lambda $ and a collection $a_ n : \xi _ n \to \eta _ n$ of morphisms of $\mathcal{F}$ lying over $R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$, such that for every $n$ the diagram

    \[ \xymatrix{ \xi _{n + 1} \ar[r]_{f_ n} \ar[d]_{a_{n + 1}} & \xi _ n \ar[d]^{a_ n} \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n } \]

    commutes.

The category of formal objects comes with a functor $\widehat{p}: \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda $ which sends an object $(R, \xi _ n, f_ n)$ to $R$ and a morphism $(R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n)$ to the map $R \to S$.

Lemma 88.7.2. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ be a category cofibered in groupoids. Then $\widehat{p} : \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda $ is a category cofibered in groupoids.

Proof. Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda $. Let $(R, \xi _ n, f_ n)$ be an object of $\widehat{\mathcal{F}}$. For each $n$ choose a pushforward $\xi _ n \to \eta _ n$ of $\xi _ n$ along $R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$. For each $n$ there exists a unique morphism $g_ n : \eta _{n + 1} \to \eta _ n$ in $\mathcal{F}$ lying over $S/\mathfrak m_ S^{n + 1} \to S/\mathfrak m_ S^ n$ such that

\[ \xymatrix{ \xi _{n + 1} \ar[d] \ar[r]_{f_ n} & \xi _ n \ar[d] \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n } \]

commutes (by the first axiom of a category cofibred in groupoids). Hence we obtain a morphism $(R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n)$ lying over $R \to S$, i.e., the first axiom of a category cofibred in groupoids holds for $\widehat{\mathcal{F}}$. To see the second axiom suppose that we have morphisms $a : (R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n)$ and $b : (R, \xi _ n, f_ n) \to (T, \theta _ n, h_ n)$ in $\widehat{\mathcal{F}}$ and a morphism $c_0 : S \to T$ in $\widehat{\mathcal{C}}_\Lambda $ such that $c_0 \circ a_0 = b_0$. By the second axiom of a category cofibred in groupoids for $\mathcal{F}$ we obtain unique maps $c_ n : \eta _ n \to \theta _ n$ lying over $S/\mathfrak m_ S^ n \to T/\mathfrak m_ T^ n$ such that $c_ n \circ a_ n = b_ n$. Setting $c = (c_ n)_{n \geq 0}$ gives the desired morphism $c : (S, \eta _ n, g_ n) \to (T, \theta _ n, h_ n)$ in $\widehat{\mathcal{F}}$ (we omit the verification that $h_ n \circ c_{n + 1} = c_ n \circ g_ n$). $\square$

Definition 88.7.3. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ be a category cofibered in groupoids. The category cofibered in groupoids $\widehat{p} : \widehat{\mathcal F} \to \widehat{\mathcal{C}}_\Lambda $ is called the completion of $\mathcal{F}$.

If $\mathcal{F}$ is a category cofibered in groupoids over $\mathcal C_\Lambda $, we have defined $\widehat{\mathcal{F}}(R)$ for $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$ in terms of the filtration of $R$ by powers of its maximal ideal. But suppose $\mathcal{I} = (I_ n)$ is a filtration of $R$ by ideals inducing the $\mathfrak {m}_ R$-adic topology. We define $\widehat{\mathcal{F}}_\mathcal {I}(R)$ to be the category with the following objects and morphisms:

  1. An object is a collection $(\xi _ n, f_ n)_{n \in \mathbf{N}}$ of objects $\xi _ n$ of $\mathcal{F}(R/I_ n)$ and morphisms $f_ n : \xi _{n + 1} \to \xi _ n$ lying over the projections $R/I_{n + 1} \to R/I_ n$.

  2. A morphism $a : (\xi _ n, f_ n) \to (\eta _ n, g_ n)$ consists of a collection $a_ n : \xi _ n \to \eta _ n$ of morphisms in $\mathcal{F}(R/I_ n)$, such that for every $n$ the diagram

    \[ \xymatrix{ \xi _{n + 1} \ar[r]^{f_ n} \ar[d]_{a_{n + 1}} & \xi _ n \ar[d]^{a_ n} \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n } \]

    commutes.

Lemma 88.7.4. In the situation above, $\widehat{\mathcal{F}}_\mathcal {I}(R)$ is equivalent to the category $\widehat{\mathcal{F}}(R)$.

Proof. An equivalence $\widehat{\mathcal{F}}_\mathcal {I}(R) \to \widehat{\mathcal{F}}(R)$ can be defined as follows. For each $n$, let $m(n)$ be the least $m$ that $I_ m \subset \mathfrak m_ R^ n$. Given an object $(\xi _ n, f_ n)$ of $\widehat{\mathcal{F}}_\mathcal {I}(R)$, let $\eta _ n$ be the pushforward of $\xi _{m(n)}$ along $R/I_{m(n)} \to R/\mathfrak m_ R^ n$. Let $g_ n : \eta _{n + 1} \to \eta _ n$ be the unique morphism of $\mathcal{F}$ lying over $R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n$ such that

\[ \xymatrix{ \xi _{m(n + 1)} \ar[rrr]_{f_{m(n)} \circ \ldots \circ f_{m(n + 1) - 1}} \ar[d] & & & \xi _{m(n)} \ar[d] \\ \eta _{n + 1} \ar[rrr]^{g_ n} & & & \eta _ n } \]

commutes (existence and uniqueness is guaranteed by the axioms of a cofibred category). The functor $\widehat{\mathcal{F}}_\mathcal {I}(R) \to \widehat{\mathcal{F}}(R)$ sends $(\xi _ n, f_ n)$ to $(R, \eta _ n, g_ n)$. We omit the verification that this is indeed an equivalence of categories. $\square$

Remark 88.7.5. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ be a category cofibered in groupoids. Suppose that for each $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$ we are given a filtration $\mathcal{I}_ R$ of $R$ by ideals. If $\mathcal{I}_ R$ induces the $\mathfrak m_ R$-adic topology on $R$ for all $R$, then one can define a category $\widehat{\mathcal{F}}_\mathcal {I}$ by mimicking the definition of $\widehat{\mathcal{F}}$. This category comes equipped with a morphism $\widehat{p}_\mathcal {I} : \widehat{\mathcal{F}}_\mathcal {I} \to \widehat{\mathcal{C}}_\Lambda $ making it into a category cofibered in groupoids such that $\widehat{\mathcal{F}}_\mathcal {I}(R)$ is isomorphic to $\widehat{\mathcal{F}}_{\mathcal{I}_ R}(R)$ as defined above. The categories cofibered in groupoids $\widehat{\mathcal{F}}_\mathcal {I}$ and $\widehat{\mathcal{F}}$ are equivalent, by using over an object $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$ the equivalence of Lemma 88.7.4.

Remark 88.7.6. Let $F: \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Identifying functors with cofibered sets, the completion of $F$ is the functor $\widehat{F} : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ given by $\widehat{F}(S) = \mathop{\mathrm{lim}}\nolimits F(S/\mathfrak {m}_ S^{n})$. This agrees with the definition in Schlessinger's paper [Sch].

Remark 88.7.7. Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda $. We claim that there is a canonical equivalence

\[ can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \longrightarrow \mathcal{F}. \]

Namely, let $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda )$ and let $(A, \xi _ n, f_ n)$ be an object of $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda }(A)$. Since $A$ is Artinian there is a minimal $m \in \mathbf{N}$ such that $\mathfrak m_ A^ m = 0$. Then $can$ sends $(A, \xi _ n, f_ n)$ to $\xi _ m$. This functor is an equivalence of categories cofibered in groupoids by Categories, Lemma 4.35.8 because it is an equivalence on all fibre categories by Lemma 88.7.4 and the fact that the $\mathfrak m_ A$-adic topology on a local Artinian ring $A$ comes from the zero ideal. We will frequently identify $\mathcal{F}$ with a full subcategory of $\widehat{\mathcal{F}}$ via a quasi-inverse to the functor $can$.

Remark 88.7.8. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda $. Then there is an induced morphism $\widehat{\varphi }: \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$ of categories cofibered in groupoids over $\widehat{\mathcal{C}}_\Lambda $. It sends an object $\xi = (R, \xi _ n, f_ n)$ of $\widehat{\mathcal{F}}$ to $(R, \varphi (\xi _ n), \varphi (f_ n))$, and it sends a morphism $(a_0 : R \to S, a_ n : \xi _ n \to \eta _ n)$ between objects $\xi $ and $\eta $ of $\widehat{\mathcal{F}}$ to $(a_0 : R \to S, \varphi (a_ n) : \varphi (\xi _ n) \to \varphi (\eta _ n))$. Finally, if $t : \varphi \to \varphi '$ is a $2$-morphism between $1$-morphisms $\varphi , \varphi ': \mathcal{F} \to \mathcal{G}$ of categories cofibred in groupoids, then we obtain a $2$-morphism $\widehat{t} : \widehat{\varphi } \to \widehat{\varphi }'$. Namely, for $\xi = (R, \xi _ n, f_ n)$ as above we set $\widehat{t}_\xi = (t_{\varphi (\xi _ n)})$. Hence completion defines a functor between $2$-categories

\[ \widehat{~ } : \text{Cof}(\mathcal{C}_\Lambda ) \longrightarrow \text{Cof}(\widehat{\mathcal{C}}_\Lambda ) \]

from the $2$-category of categories cofibred in groupoids over $\mathcal{C}_\Lambda $ to the $2$-category of categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda $.

Remark 88.7.9. We claim the completion functor of Remark 88.7.8 and the restriction functor $|_{\mathcal{C}_\Lambda } : \text{Cof}(\widehat{\mathcal{C}}_\Lambda ) \to \text{Cof}(\mathcal{C}_\Lambda )$ of Remarks 88.5.2 (15) are “2-adjoint” in the following precise sense. Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\text{Cof}(\mathcal{C}_\Lambda ))$ and let $\mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\text{Cof}(\widehat{\mathcal{C}}_\Lambda ))$. Then there is an equivalence of categories

\[ \Phi : \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \mathcal{G}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(\mathcal{G}, \widehat{\mathcal{F}}) \]

To describe this equivalence, we define canonical morphisms $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda }}$ and $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ as follows

  1. Let $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda ))$ and let $\xi $ be an object of the fiber category $\mathcal{G}(R)$. Choose a pushforward $\xi \to \xi _ n$ of $\xi $ to $R/\mathfrak m_ R^ n$ for each $n \in \mathbf{N}$, and let $f_ n : \xi _{n + 1} \to \xi _ n$ be the induced morphism. Then $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda }}$ sends $\xi $ to $(R, \xi _ n, f_ n)$.

  2. This is the equivalence $can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ of Remark 88.7.7.

Having said this, the equivalence $\Phi : \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \mathcal{G}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \to \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(\mathcal{G}, \widehat{\mathcal{F}})$ sends a morphism $\varphi : \mathcal{G}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ to

\[ \mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda }} \xrightarrow {\widehat{\varphi }} \widehat{\mathcal{F}} \]

There is a quasi-inverse $\Psi : \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \mathcal{G}, \widehat{\mathcal{F}}) \to \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \mathcal{G}|_{\mathcal{C}_\Lambda }, \mathcal{F})$ to $\Phi $ which sends $\psi : \mathcal{G} \to \widehat{\mathcal{F}}$ to

\[ \mathcal{G}|_{\mathcal{C}_\Lambda } \xrightarrow {\psi |_{\mathcal{C}_\Lambda }} \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \to \mathcal{F}. \]

We omit the verification that $\Phi $ and $\Psi $ are quasi-inverse. We also do not address functoriality of $\Phi $ (because it would lead into 3-category territory which we want to avoid at all cost).

Remark 88.7.10. For a category $\mathcal{C}$ we denote by $\text{CofSet}(\mathcal{C})$ the category of cofibered sets over $\mathcal{C}$. It is a $1$-category isomorphic the category of functors $\mathcal{C} \to \textit{Sets}$. See Remarks 88.5.2 (11). The completion and restriction functors restrict to functors $\widehat{~ } : \text{CofSet}(\mathcal{C}_\Lambda ) \to \text{CofSet}(\widehat{\mathcal{C}}_\Lambda )$ and $|_{\mathcal{C}_\Lambda } : \text{CofSet}(\widehat{\mathcal{C}}_\Lambda ) \to \text{CofSet}(\mathcal{C}_\Lambda )$ which we denote by the same symbols. As functors on the categories of cofibered sets, completion and restriction are adjoints in the usual 1-categorical sense: the same construction as in Remark 88.7.9 defines a functorial bijection

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(G|_{\mathcal{C}_\Lambda }, F) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(G, \widehat{F}) \]

for $F \in \mathop{\mathrm{Ob}}\nolimits (\text{CofSet}(\mathcal{C}_\Lambda ))$ and $G \in \mathop{\mathrm{Ob}}\nolimits (\text{CofSet}(\widehat{\mathcal{C}}_\Lambda ))$. Again the map $\widehat{F}|_{\mathcal{C}_\Lambda } \to F$ is an isomorphism.

Remark 88.7.11. Let $G : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ be a functor that commutes with limits. Then the map $G \to \widehat{G|_{\mathcal{C}_\Lambda }}$ described in Remark 88.7.9 is an isomorphism. Indeed, if $S$ is an object of $\widehat{\mathcal{C}}_\Lambda $, then we have canonical bijections

\[ \widehat{G|_{\mathcal{C}_\Lambda }}(S) = \mathop{\mathrm{lim}}\nolimits _ n G(S/\mathfrak {m}_ S^ n) = G(\mathop{\mathrm{lim}}\nolimits _ n S/\mathfrak {m}_ S^ n) = G(S). \]

In particular, if $R$ is an object of $\widehat{\mathcal{C}}_\Lambda $ then $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda }}$ because the representable functor $\underline{R}$ commutes with limits by definition of limits.

Remark 88.7.12. Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda $. It defines a functor $\underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ as described in Remarks 88.5.2 (12). As usual we identify this functor with the associated cofibered set. If $\mathcal{F}$ is a cofibered category over $\mathcal{C}_\Lambda $, then there is an equivalence of categories

88.7.12.1
\begin{equation} \label{formal-defos-equation-formal-objects-maps} \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \underline{R}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \longrightarrow \widehat{\mathcal{F}}(R). \end{equation}

It is given by the composition

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \underline{R}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \xrightarrow {\Phi } \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow {\sim } \widehat{\mathcal{F}}(R) \]

where $\Phi $ is as in Remark 88.7.9 and the second equivalence comes from the 2-Yoneda lemma (the cofibered analogue of Categories, Lemma 4.41.1). Explicitly, the equivalence sends a morphism $\varphi : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ to the formal object $(R, \varphi (R \to R/\mathfrak {m}_ R^ n), \varphi (f_ n))$ in $\widehat{\mathcal{F}}(R)$, where $f_ n : R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n$ is the projection.

Assume a choice of pushforwards for $\mathcal{F}$ has been made. Given any $\xi \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{F}}(R))$ we construct an explicit $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ which maps to $\xi $ under (88.7.12.1). Namely, say $\xi = (R, \xi _ n, f_ n)$. An object $\alpha $ in $\underline{R}|_{\mathcal{C}_\Lambda }$ is the same thing as a morphism $\alpha : R \to A$ of $\widehat{\mathcal{C}}_\Lambda $ with $A$ Artinian. Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_ A^ m = 0$. Then $\alpha $ factors through a unique $\alpha _ m : R/\mathfrak m_ R^ m \to A$ and we can set $\underline{\xi }(\alpha ) = \alpha _{m, *}\xi _ m$. We omit the description of $\underline{\xi }$ on morphisms and we omit the proof that $\underline{\xi }$ maps to $\xi $ via (88.7.12.1).

Assume a choice of pushforwards for $\widehat{\mathcal{F}}$ has been made. In this case the proof of Categories, Lemma 4.41.1 gives an explicit quasi-inverse

\[ \iota : \widehat{\mathcal{F}}(R) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \underline{R}, \widehat{\mathcal{F}}) \]

to the 2-Yoneda equivalence which takes $\xi $ to the morphism $\iota (\xi ) : \underline{R} \to \widehat{\mathcal{F}}$ sending $f \in \underline{R}(S) = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(R, S)$ to $f_*\xi $. A quasi-inverse to (88.7.12.1) is then

\[ \widehat{\mathcal{F}}(R) \xrightarrow {\iota } \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow {\Psi } \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \underline{R}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \]

where $\Psi $ is as in Remark 88.7.9. Given $\xi \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{F}}(R))$ we have $\Psi (\iota (\xi )) \cong \underline{\xi }$ where $\underline{\xi }$ is as in the previous paragraph, because both are mapped to $\xi $ under the equivalence of categories (88.7.12.1). Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda }}$ (see Remark 88.7.11) and unwinding the definitions of $\Phi $ and $\Psi $ we conclude that $\iota (\xi )$ is isomorphic to the completion of $\underline{\xi }$.

Remark 88.7.13. Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda $. Let $\xi = (R, \xi _ n, f_ n)$ and $\eta = (S, \eta _ n, g_ n)$ be formal objects of $\mathcal{F}$. Let $a = (a_ n) : \xi \to \eta $ be a morphism of formal objects, i.e., a morphism of $\widehat{\mathcal{F}}$. Let $f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in $\widehat{\mathcal{C}}_\Lambda $. Then we obtain a $2$-commutative diagram

\[ \xymatrix{ \underline{R}|_{\mathcal{C}_\Lambda } \ar[rd]_{\underline{\xi }} & & \underline{S}|_{\mathcal{C}_\Lambda } \ar[ll]^ f \ar[ld]^{\underline{\eta }} \\ & \mathcal{F} } \]

where $\underline{\xi }$ and $\underline{\eta }$ are the morphisms constructed in Remark 88.7.12. To see this let $\alpha : S \to A$ be an object of $\underline{S}|_{\mathcal{C}_\Lambda }$ (see loc. cit.). Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_ A^ m = 0$. We get a commutative diagram

\[ \xymatrix{ R \ar[d]^ f \ar[r] & R/\mathfrak m_ R^ m \ar[d]_{f_ m} \ar[rd]^{\beta _ m} \\ S \ar[r] & S/\mathfrak m_ S^ m \ar[r]^{\alpha _ m} & A } \]

such that the bottom arrows compose to give $\alpha $. Then $\underline{\eta }(\alpha ) = \alpha _{m, *}\eta _ m$ and $\underline{\xi }(\alpha \circ f) = \beta _{m, *}\xi _ m$. The morphism $a_ m : \xi _ m \to \eta _ m$ lies over $f_ m$ hence we obtain a canonical morphism

\[ \underline{\xi }(\alpha \circ f) = \beta _{m, *}\xi _ m \longrightarrow \underline{\eta }(\alpha ) = \alpha _{m, *}\eta _ m \]

lying over $\text{id}_ A$ such that

\[ \xymatrix{ \xi _ m \ar[r] \ar[d]^{a_ m} & \beta _{m, *}\xi _ m \ar[d] \\ \eta _ m \ar[r] & \alpha _{m, *}\eta _ m } \]

commutes by the axioms of a category cofibred in groupoids. This defines a transformation of functors $\underline{\xi } \circ f \to \underline{\eta }$ which witnesses the 2-commutativity of the first diagram of this remark.

Remark 88.7.14. According to Remark 88.7.12, giving a formal object $\xi $ of $\mathcal{F}$ is equivalent to giving a prorepresentable functor $U : \mathcal{C}_\Lambda \to \textit{Sets}$ and a morphism $U \to \mathcal{F}$.


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